INTERPRETING A FIELD IN ITS HEISENBERG GROUP
2022
We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by
$H(F)$
the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in
$H(F)$
, using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in
$H(F)$
using computable
$\Sigma _1$
formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in
$H(F)$
of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in
$H(F)$
. Looking at what was used to arrive at this parameter-free interpretation of F in
$H(F)$
, we give general conditions sufficient to eliminate parameters from interpretations.
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